Abstract

We calculate the perturbative value of the free energy in Lattice QCD,
up to three loops. Our calculation is performed using Wilson gluons
and the Sheikholeslami - Wolhert
(clover) improved action for fermions.

The free energy is directly related to the average plaquette.
To carry out the calculation, we compute all relevant Feynman diagrams
up to 3 loops, using a set of automated procedures in Mathematica;
numerical evaluation of the resulting loop integrals is performed on
finite lattice, with subsequent extrapolation to infinite size.

The results are presented as a function of the fermion mass m, for
any SU(Nc) gauge group, and for an arbitrary number of fermion flavors.
In order to enable independent comparisons,
we also provide the results on a per diagram basis,
for a specific mass value.

I Formulation of the Problem

In this work we calculate the free energy of QCD on the lattice,
up to three loops in perturbation theory.
We employ Wilson gluons and the O(a) improved Sheikholeslami-Wohlert
(clover) [1] action for fermions.
The purpose of this action is to reduce finite lattice spacing effects,
leading to a faster approach to the continuum. Dynamical
simulations employing the clover action are currently in progress by the
CP-PACS/JLQCD [2] and UKQCDSF [3] collaborations and therefore perturbative studies of
properties of the QCD action with clover quarks are worthy of being
undertaken. The free energy, in the simpler case of
Wilson fermions, was studied in [4].

The free energy in QCD on the lattice can be related to the average plaquette.
The results find several applications, for example:
a) In improved scaling schemes, using an appropriately defined
effective coupling which depends on the average plaquette (see, e. g.,
[5, 6]),
b) In long standing efforts, starting with [7], to determine the
value of the gluon condensate,
c) In studies of the interquark potential [8], and
d) As a test of perturbation theory, at its limits of applicability.

Indeed, regarding point (d) above, the plaquette expectation value is
a prototype for additive renormalization of a composite, dimensionful
operator. The vacuum diagrams contributing to such a calculation are
power divergent in the lattice spacing and may well dominate over any
nonperturbative signal in a numerical simulation.

Starting from the Wilson formulation of QCD on the lattice,
with the addition of the clover (SW) fermion term, the action reads in
standard notation:

SL

≡

SG+SF,

SG

=

1g2∑x,μ,νTr[1−Uμ,ν(x)],

SF

=

∑f∑x(4r+mB)¯ψf(x)ψf(x)

(1)

−12∑f∑x,μ[¯ψf(x)(r−γμ)Uμ(x)ψf(x+^μ)+¯ψf(x+^μ)(r+γμ)Uμ(x)†ψf(x)]

+i4cSW∑f∑x,μ,ν¯ψf(x)σμν^Fμν(x)ψf(x)

where:^Fμν≡18(Qμν−Qνμ),Qμν=Uμ,ν+Uν,−μ+U−μ,−ν+U−ν,μ

(2)

Here Uμ,ν(x) is the usual product of SU(Nc) link variables
Uμ(x) along the perimeter of a plaquette in the μ-ν
directions, originating at x;
g denotes the bare coupling constant; r is the Wilson parameter,
which will be assigned its standard value r=1;
f is a flavor index; σμν=(i/2)[γμ,γν].
Powers of the lattice spacing a have been omitted and
may be directly reinserted by dimensional counting.

The clover coefficient cSW is a free parameter for the
purposes of the present calculation and our results will be presented
as a polynomial in cSW , with coefficients which we
compute. Preferred values for cSW have been suggested by both
perturbative (1-loop) [1] and non-perturbative [9] studies.

We use the standard covariant gauge-fixing term [10]; in terms of
the vector field Qμ(x)[Uμ(x)=exp(igQμ(x))], it
reads:

Sgf=λ0∑μ,ν∑xTrΔ−μQμ(x)Δ−νQν(x),Δ−μQν(x)≡Qν(x−^μ)−Qν(x)

(3)

Having to compute a gauge invariant quantity, we chose to work
in the Feynman gauge, λ0=1.
Covariant gauge fixing produces the following
action for the ghost fields ω and ¯¯¯ω

Δ+μω(x)+ig0[Qμ(x),ω(x)]+ig02[Qμ(x),Δ+μω(x)]

(4)

−g2012[Qμ(x),[Qμ(x),Δ+μω(x)]]

−g40720[Qμ(x),[Qμ(x),[Qμ(x),[Qμ(x),Δ+μω(x)]]]]+⋯)},

Δ+μω(x)≡ω(x+^μ)−ω(x)

Finally the change of integration variables from links to vector
fields yields a jacobian that can be rewritten as
the usual measure term Sm in the action:

(5)

In Sgh and Sm we have written out only
terms relevant to our computation.
The full action is:

S=SL+Sgf+Sgh+Sm

(6)

The average value of the action density, S/V, is directly related to
the average plaquette. For the gluon part we have:

⟨SG/V⟩=6βEG,EG≡1−1NcTr⟨Uμ,ν(x)⟩,β=2Nc/g2

(7)

As for ⟨SF/V⟩, it is trivial in any action which is
bilinear in the fermion fields, and leads to:

⟨SF/V⟩=−4NcNf

(8)

(Nf : number of fermion flavors).

We will calculate EG in perturbation theory:

EG=c1g2+c2g4+c3g6+⋯

(9)

The n-loop coefficient can be written as cn=cGn+cFn where
cGn is the contribution of diagrams without fermion loops and
cFn comes from diagrams containing fermions. The coefficients
cGn have been known for some time up to 3
loops [11] (also in 3 dimensions [12], where they are
applied to “Magnetostatic” QCD [13] and to dimensionally
reduced QCD [14, 15]). Independent estimates of higher loop
coefficients have also been obtained using stochastic perturbation
theory [16]. The fermionic coefficients cFn are known to
2 loops for overlap fermions [17] and up to 3
loops for Wilson fermions [4]; in the present work we extend
this computation to the clover action.

The calculation of cn proceeds most conveniently by computing first the free energy −(lnZ)/V, where Z is the full partition function:

Z≡∫[DUD¯ψiDψi]exp(−S)

(10)

Then, EG is extracted through

EG=−16∂∂β(lnZV)

(11)

In particular, the perturbative expansion of (lnZ)/V :

(lnZ)/V=d0−3(N2c−1)2lnβ+d1β+d2β2+⋯

(12)

leads immediately to the relations:

c2=d1/(24N2c),c3=d2/(24N3c)

(13)

Ii Calculation and Results

A Total of 62 Feynman diagrams contribute to the present calculation,
up to three loops. The first 36 diagrams are totally gluonic, and the
others have both gluon and fermion contributions; these are shown
in Appendix A. The involved algebra of lattice
perturbation theory was carried out using our computer
package in Mathematica. The value for each diagram is computed numerically for a
sequence of finite lattices, with typical size L≤36.

Certain diagrams must be grouped into infrared-finite
sets, before extrapolating their values to infinite lattice size
(diagrams 11+12+13, 22 through 36, 43+53, 44+52+58, 46+56, 51+57,
55+60, 61+62). Extrapolation leads to a (small) systematic error, which
is estimated quite accurately; a consise description of the procedure
is provided in Ref. [12].

Diagrams in the shape of a triangular pyramid (18, 19, 20, 49, 50) are the most CPU
demanding, since integration over the 3 loop momenta cannot be
factorized; these diagrams were necessarily evaluated for smaller L,
but fortunately L∼16 was already sufficient for a very stable
extrapolation in these cases. Diagram 40 vanishes identically by color antisymmetry.

The coefficients h2,h30,h31,h32 depend polynomially on
the clover parameter cSW:

h2=h(0)2+h(1)2cSW+h(2)2c2SW

(16)

h3i=h(0)3i+h(1)3icSW+h(2)3ic2SW+h(3)3ic3SW+h(4)3ic4SW

We have calculated h(j)2, h(j)3i
for typical values of the Lagrangian (unrenormalized) fermion mass
parameter m, which is connected to the familiar
hopping expansion parameter κ=1/(2m+8). Our results are listed in Appendix B. A complete per diagram breakdown of the results would be far too lengthy to
present; instead, for potential comparisons, we provide in Appendix B
a breakdown only for a particular value of m.

In Figs. 1 and 2 we present the dependence of c2 and c3 ,
respectively, on m, for three typical values of cSW.

Fig. 1. The dependence of c2 on the fermion mass m,
for some standard values of cSW . Nc=3, Nf=2.

Fig. 2. The dependence of c3 on the fermion mass m,
for some standard values of cSW . Nc=3, Nf=2.

We list below some examples of values for EG , setting Nc=3.
For Nf=0 we have:

EG=(1/3)g2+0.0339109931(3)g4+0.0137063(2)g6

(17)

For two degenerate flavors (Nf=2) and
m=−0.518106 (corresponding to κ=(8+2m)−1=0.1436):

It is seen that the 3-loop coefficients are quite pronounced for
typical values of g used in numerical simulations. For convenience,
the behaviour of EG versus β is also presented in Fig. 3, for
the same parameter values as in Eqs. (18,19).

Fig. 3. EG as a function
of β, for
Nc=3, Nf=2, and specific mass values.

The detailed results, tabulated in Appendix B for arbitrary values of
Nc, Nf, cSW, show a very smooth behaviour as a function
of m; consequently, one is able to reconstruct EG also for
arbitrary values of m by naive interpolation, to excellent
precision.

Appendix A

Fig. 4. Feynman diagrams contributing to the free
energy, at one, two, and three loops.

Figure 4 depicts all diagrams contributing to the free energy at 1
loop (diagram 1), 2 loops (2-6, 37-38), and 3 loops (7-36, 39-62). Solid (curly, dashed)
lines represent fermions (gluons, ghosts), and the filled square is the contribution
from the measure part of the action. The filled circle, corresponding
to the non-fermionic part of the 1-loop gluon self-energy, is given in
Figure 5.

Appendix B

Tables I-IV provide a per diagram breakdown of our results, at a
given value of m (m=0.038), in order to allow for potential
comparisons and cross checks. The total results for the coefficients
h(j)2, h(j)30, h(j)31, h(j)32 are listed
in Tables V-VIII, respectively, for a wide selection of m values
which are used in the literature.
Given the smooth dependence of all these coefficients on m,
interpolations to other intermediate values of m can be performed
with great accuracy.